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Theorem dia1eldmN 31231
Description: The fiducial hyperplane (largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
dia1eldm.h  |-  H  =  ( LHyp `  K
)
dia1eldm.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
dia1eldmN  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  dom  I
)

Proof of Theorem dia1eldmN
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 dia1eldm.h . . . 4  |-  H  =  ( LHyp `  K
)
31, 2lhpbase 30187 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
43adantl 452 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  ( Base `  K ) )
5 hllat 29553 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
6 eqid 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
71, 6latref 14159 . . 3  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W ( le `  K ) W )
85, 3, 7syl2an 463 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W ( le `  K ) W )
9 dia1eldm.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
101, 6, 2, 9diaeldm 31226 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( W  e.  dom  I 
<->  ( W  e.  (
Base `  K )  /\  W ( le `  K ) W ) ) )
114, 8, 10mpbir2and 888 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  dom  I
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   dom cdm 4689   ` cfv 5255   Basecbs 13148   lecple 13215   Latclat 14151   HLchlt 29540   LHypclh 30173   DIsoAcdia 31218
This theorem is referenced by:  dia1elN  31244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-poset 14080  df-lat 14152  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-disoa 31219
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